Editing Euclidean vector (section)

===Generalizations===
In physics, as well as mathematics, a vector is often identified with a [[tuple]] of components, or list of numbers, that act as scalar coefficients for a set of [[basis vector]]s. When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate. The vector is called ''covariant'' or ''contravariant'', depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as [[gradient]]. If you change units (a special case of a [[change of basis]]) from meters to millimeters, a scale factor of 1/1000, a displacement of 1&nbsp;m becomes 1000&nbsp;mm—a contravariant change in numerical value. In contrast, a gradient of 1&nbsp;[[Kelvin|K]]/m becomes 0.001&nbsp;K/mm—a covariant change in value (for more, see [[covariance and contravariance of vectors]]). [[Tensor]]s are another type of quantity that behave in this way; a vector is one type of [[tensor]].

In pure [[mathematics]], a vector is any element of a [[vector space]] over some [[field (mathematics)|field]] and is often represented as a [[coordinate vector]]. The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".